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A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

%I A327615 #31 Jan 11 2021 15:12:38
%S A327615 1,1,1,3,2,1,1,5,8,6,2,1,1,8,19,25,16,7,2,1,1,11,40,73,73,47,19,7,2,1,
%T A327615 1,15,77,194,263,232,133,58,20,7,2,1,1,19,132,454,835,951,719,397,164,
%U A327615 61,20,7,2,1,1,24,217,984,2385,3507,3365,2306,1177,490,175,62,20,7,2,1
%N A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.
%C A327615 Covering k vertices means there are no vertices of degree zero.
%H A327615 Andrew Howroyd, <a href="/A327615/b327615.txt">Table of n, a(n) for n = 1..650</a> (rows 1..25)
%F A327615 T(n,k) = A290428(n,k) - A290428(n,k-1).
%e A327615 Triangle begins:
%e A327615   1,  1;
%e A327615   1,  3,   2,   1;
%e A327615   1,  5,   8,   6,   2,   1;
%e A327615   1,  8,  19,  25,  16,   7,   2,   1;
%e A327615   1, 11,  40,  73,  73,  47,  19,   7,   2,  1;
%e A327615   1, 15,  77, 194, 263, 232, 133,  58,  20,  7,  2, 1;
%e A327615   1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1;
%e A327615   ...
%o A327615 (PARI)
%o A327615 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A327615 edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
%o A327615 C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
%o A327615 T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
%o A327615 { my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }
%Y A327615 Row sums are A007717.
%Y A327615 Columns k=2..3 are A024206, A327728.
%Y A327615 Cf. A136564, A290428, A309936, A322115.
%K A327615 nonn,tabf
%O A327615 1,4
%A A327615 _Andrew Howroyd_, Oct 23 2019